Adaptive Strategies for Transport Equations

This paper is concerned with a posteriori error bounds for linear transport equations and related questions of contriving corresponding adaptive solution strategies in the context of Discontinuous Petrov Galerkin schemes. After indicating our motivation for this investigation in a wider context the first major part of the paper is devoted to the derivation and analysis of a posteriori error bounds that, under mild conditions on variable convection fields, are efficient and, modulo a data-oscillation term, reliable. In particular, it is shown that these error estimators are computed at a cost that stays uniformly proportional to the problem size. The remaining part of the paper is then concerned with the question whether typical bulk criteria known from adaptive strategies for elliptic problems entail a fixed error reduction rate also in the context of transport equations. This turns out to be significantly more difficult than for elliptic problems and at this point we can give a complete affirmative answer for a single spatial dimension. For the general multidimensional case we provide partial results which we find of interest in their own right. An essential distinction from known concepts is that global arguments enter the issue of error reduction. An important ingredient of the underlying analysis, which is perhaps interesting in its own right, is to relate the derived error indicators to the residuals that naturally arise in related least squares formulations. This reveals a close interrelation between both settings regarding error reduction in the context of adaptive refinements

On the stability of DPG formulations of transport equations

In this paper we formulate and analyze a Discontinuous Petrov- Galerkin formulation of linear transport equations with variable convection fields. We show that a corresponding infinite dimensional mesh-dependent variational formulation, in which besides the principal field its trace on the mesh skeleton is also an unknown, is uniformly stable with respect to the mesh, where the test space is a certain product space over the underlying domain partition. Our main result then states the following. For piecewise polynomial trial spaces of degree m, we show under mild assumptions on the convection field that piecewise polynomial test spaces of degree m+1 over a refinement of the primal partition with uniformly bounded refinement depth give rise to uniformly (with respect to the mesh size) stable Petrov-Galerkin discretizations. The partitions are required to be shape regular but need not be quasi-uniform. An important startup ingredient is that for a constant convection field one can identify the exact optimal test functions with respect to a suitably modified but uniformly equivalent broken test space norm as piecewise polynomials. These test functions are then varied towards simpler and stably computable near-optimal test functions for which the above result is derived via a perturbation analysis. We conclude indicating some consequences of the results that will be treated in forthcoming work

Smooth surface interpolation to scattered data using interpolatory subdivision algorithms

10.1016/0898-1221(96)00115-0Computers and Mathematics with Applications32393-110CMAP